Another cool thing I learned along my journey, is that there is a function that creates a Cartesian parabola. I had no idea this existed, but Descartes' used it as one of his 'simple curves'. To better understand this, I created an applet that helped me create a Cartesian parabola.
Move n, a, and b to see how they move the parabola and the line.
Move D_1 to move point D to see the locus of all possible values of C1 and C2 outline the cartesian parabola.
The constant, 1/n is leading coefficient of function y.
Point E is a units away from point D on the y-axis.
Point A is a point on the x-axis. Line f goes through the points A and E.
Line f crosses parabola y at two points, C1 and C2.
The purple dashed line is the locus of all possible values of C1 and C2.
We can see it outlines the cartesian parabola.
(Rubinstein, 5)